The performance of sailing yachts in oblique seas

Model of 1967 Intrepid tn oblique seas

The Performance of Sailing Yachts in Oblique Seas

David R. Pedrick

The difference in the effects of rough water on similar sailing yachts has been one of the intriguing puzzles that sailors, designers, and researchers have long tried to understand. It is not uncommon for t w o yachts of equal performance in smooth-sea conditions to have their speed or pointing ability reduced by different amounts when encountering waves. To investigate the causes of such behavior, it is important to have a rational procedure to analyze how changes in hull form, weight distribution, rig, and other design features affect the speed and motions of sailing yachts. This paper discusses the relationship of wind to rough water and of motions and added resistance to wave length and height. It then describes a procedure to predict motions, sailing speed, and speed-made-good to windward in realistic windward sailing conditions. The procedure utilizes results of heeled and yawed model tests of 12-metre yachts in oblique regular waves to predict performance in a Pierson-Mosl<owitz sea state corresponding closely to the equilibrium true wind speed.

I n t r o d u c t i o n

One o f the most valuable tools i n the development of yacht design has been the model tank. P r i o r t o 1932, model tests were l i m i t e d to simple u p r i g h t tests, and although yachts were generally improved by such tests i t was clear t h a t f u r t h e r i n f o r m a t i o n about w i n d w a r d performance was needed. D a v i d -son undertook the investigation of heeled and yawed tests, thereby simulating the stability and side force required to counteract the forces generated by the sails. I n reference [1]^ he described the development of this side force technique. As 1 Chief, Scientific Section, Sparkman & Stephens, Inc., New York, N . Y .

^ Numbers in brackets designate References at end of paper. ' Presented at the Chesapeake Sailing Yacht Symposium, Annapolis, Maryland, January 19,1974.

an example, he compared two 6-metre yachts, Jack and Jill. Jack, the newer boat, was f a i r l y good d o w n w i n d , b u t signifi-cantly slower t h a n Jill to w i n d w a r d . Figures 1 and 2, taken f r o m [1], show Davidson's results. Indeed, Jack had better up-r i g h t up-resistances t h a n Jill below about 6 knots boat speed. However, Jill's tests proved to be several percent faster to

w i n d w a r d . I f the designer had only k n o w n beforehand. . . ! Davidson recognized t h a t the effects of encountered waves are an i m p o r t a n t consideration, b u t he hypothesized t h a t yachts are affected nearly equally, and t h a t his more immedi-ate questions of stability, heel, and leeway could be separimmedi-ated f r o m rough-water effects. The s i m p l i f i c a t i o n o f using smooth-water heeled tests continues to serve w e l l i n the development of b e t t e r - p e r f o r m i n g sailing yachts. However, i t is an inter-esting and p o t e n t i a l l y f r u i t f u l endeavor to take another step

30O, 250 200 0»

I

Boat Speed, knots

Fig. 1 Comparison of uprigtit resistances of 6-metre boats Jack and Jill

toward s i m u l a t i n g realistic sailing conditions by including tbe effects t h a t rough water has on boat speed and sailing angle. Spens et al. [2] reported on an investigation o f the 12-metre yacht Sovereign i n oblique seas, showing the influence of l o n -g i t u d i n a l wei-ght d i s t r i b u t i o n on motions, resistance, and speed loss. T h e present paper shows how the results of side force testing i n oblique regular waves can be used to make predictions of motions and speed to windward, accounting f o r both added resistance and side force effects i n any specified sea condition. T o understand the procedure used i n the rough-water predictions, however, i t is f o r s t i m p o r t a n t to be f a m i l i a r w i t h smooth-water testing and speed predictions.

S m o o t h - w a t e r t e s t p r e d i c t i o n s

Components of resistance. Prediction of the performance of full-sized saihng yachts f r o m small models is made possible by separating the t o t a l resistance into its various components and expanding the results according to appropriate scaling laws. T h e simplest case is u p r i g h t testing at zero yaw, or lee-way. Here the model has symmetrical f l o w , p o r t and star-board, and no side force is generated. T h e t o t a l resistance can be divided i n t o skin f r i c t i o n and wavemaking. A c t u a l l y , i n ad-d i t i o n t o wavemaking, there are effects such as f o r m ad-drag anad-d eddy-making t h a t are l u m p e d together i n the second compo-nent, and i t is therefore called residuary resistance^:

Rr = R, + R, ₍₁₎

W i l l i a m Froude established this principle i n the late 1800's. He recognized t h a t a model and a ship moving at speeds p r o -p o r t i o n a l to the square root of their res-pective waterline

m O s 4) ^ . CO

%

ho'

jj

f r

o'

Speed, knots

Comparison of calculated close-hauled speeds of 6-metre boats Jack and Jill

lengths had geometrically similar wave patterns, and t h a t the wavemaking resistance was p r o p o r t i o n a l to t h e i r displace-ments. T h i s is now known as Froude scaling, and the speed re-lationship is expressed as either t h e speed/length ratio V/VLUJI, or the Froude number:

(2)

F r i c t i o n , however, is related to an entirely d i f f e r e n t phe-nomenon. F r i c t i o n of water along the h u l l results i n the g r o w t h o f a boundary layer of water t h a t drags alongside and astern of the h u l l . I t is therefore a f u n c t i o n of the viscosity of water (fresh or salt, and at a particular temperature), the length of the h u l l , and the speed at w h i c h i t is traveling. T h i s relationship was established b y Osborne Reynolds and is called the Reynolds number:

Rn = vl

(3) He determined that the coefficient o f f r i c t i o n *

Rf C , =

3 These components of drag, although used for many years, have shortcomings described in [3].

could be directly related to this number. However, i t is f u r t h e r complicated by large differences between laminar and t u r b u lent f l o w . I n the testing of small models, turbulence is a r t i f i -cially induced so t h a t t u r b u l e n t f r i c t i o n is assured. As a simp l i f i c a t i o n , i t has n o r m a l l y been assumed t h a t the yacht's f r i c -t i o n a l drag is equivalen-t -to -the drag of a f l a -t p l a n k having -the same length and surface area as the yacht to her test water-line. (See Fig. 3.)

Because of the coefficient f o r m used f o r relating f r i c t i o n of the m o d e l and ship, i t is convenient to use coefficients f o r other resistance terms. I t can be shown t h a t scaling i n propor-t i o n propor-t o (VzpSu'^) is equivalenpropor-t propor-to scaling i n p r o p o r propor-t i o n propor-to dis-placement, or Froude scaling.^

W h e n the yacht is heeled, t w o more components of resis-tance come i n t o play. F i r s t , due t o the d i f f e r e n t shape t h a t the

Force coefficients are defined as the force divided hy (^-zpSu'^). ^ For further discussion of sailing yacht model expansion, see refer-ence [4].

water sees, there is a change i n resistance f r o m the u p r i g h t case. Assuming zero side force, which occurs close to zero lee-way, there is generally an increase of up to about 5 percent f o r slender hulls and about 12 percent f o r beamy hulls at 30-deg heel. T h e n , as side force is generated by a combination of lee-way angle and f o r w a r d speed, there is a component of induced drag t h a t is the inevitable result of producing side force. I t is nearly a constant f u n c t i o n of (side force)^ or ( l i f t ) ^ for a par-ticular speed and heel angle. T h e slope of a line of resistance versus ( l i f t ) ^ yields a "drag slope coefficient" (Fig. 4):

T h e induced drag can then be f o u n d by the equation

Thus, f o r a given speed and side force, the t o t a l w i n d w a r d re-sistance can be w r i t t e n as

(4)

+

Values o f (RQ/RU) and (dCR/sCL^) for a yacht at a particular heel angle can be f o u n d by averaging the values of several speeds at that heel angle, or by p l o t t i n g the values against speed (see reference [5]).

Side force. I t has generally been beheved t h a t b o t h side force and its corresponding induced drag and leeway angle are independent of Reynolds number effects. Therefore, they are expanded using Froude scaling (constant coefficient). (See Fig. 5.)

Other properties of side force are t h a t i t is nearly directly

proportional to leeway angle, holding speed constant, and t h a t for a given leeway i t is nearly p r o p o r t i o n a l to (speed)^. Due to the hull's assymetry at zero yaw, there is usually some small amount of side force i n t h a t condition. T h i s can be treated by considering a small yaw angle offset 8 f o r zero side force.

F r o m Fig. 6 i t is seen t h a t l i f t can be expressed i n terms of a lift-slope coefficient (dCJda), where a is the angle of attack.

Fig. 4 Plot of total resistance versus (lift)^ at constant speed

SAIL FORCES H

Fig. 3 Friction lines in the laminar-turbulent region

HULL FORCES

Fig. 5 Forces and couples for equilibrium of heeled sailing yacht

-Nomenclature-c = wave celerity added resistance due to waves z = heaving motion or amplitude

= wave energy per unit area of sea surface Rl = induced drag due to l i f t a = angle of attack

Fn Froude number: v/VgLwi Ro resistance at zero side force /? = apparent wind angle to ship's head

g acceleration due to gravity RT = _{total resistance y = true wind angle to ship's course, sailing}

H = side force Ru = upright resistance angle

h = height from center of pressure of hull to RAO = response amplitude operator 5 = leeway angle offset center of effort of sails Rn = Reynolds number: ul/v É = l i f t slope increment wave height or double amplitude, crest RW = subscript for rough water = surface-wave amplitude

to trough S = _{wetted surface angle of pitch}

k wave number: I-K jhw • oj'^/g S = _{spectral density (with subscript) = angle between wave direction and}

kyy longitudinal radius of gyration SW = subscript for smooth water ship's course

L = _{l i f t} Tu, = period of a wave V = coefficient of kinematic viscosity

Lw = _{length of wave: that is, crest to crest} V. Vs = _{ship speed, knots p} = mass density Lull = _{length on load waterline ,} VA = apparent wind speed, knots <p angle of heel

I = characteristic length VMG = speed-made-good to windward, knots 'p = _{angle of yaw, leeway angle}

m mass (of yacht) VT = true wind speed, knots w = circular frequency of wave: iir/Tw

nio = _{variance of spectrum} V = ship speed, We = frequency of encounter

T h e hft-slope coefficient is f a i r l y constant for a yacht at a given heel angle and over a range of w i n d w a r d speeds:

a = \p + 5

dC,

L = -Q^ii' + 8){\pSv') (7) I n the coefficient, is used to approximate the area of the

f o i l , rather t h a n using the t o t a l surface o f b o t h sides. I t should be understood t h a t l i f t is defined as the force component nor-mal to the yacht's vertical axis and her course. The component of this force parallel to the water surface is called side force, H = L coscp.

Conditions for e q u i l i b r i u m . I n model testing and perfor-mance prediction i t is convenient to assume a heel angle and speed and then determine the forces, w i n d speeds, and angles t h a t correspond to steady-state operation. Sailing e q u i l i b r i u m is attained when:

1. The l i f t generated by the h u l l is equal and opposite to t h a t generated by the sails.

2. The corresponding heeling moment couple is equal and opposite t o the yacht's righting m o m e n t couple at the given heel angle.

3. T h e d r i v i n g force is equal and opposite to the resistance at the given speed.

I t has been f o u n d t h a t a yacht's r i g h t i n g moment couple is

LEEWAY, y/

Fig. 6 Lift coefficient versus leeway angle at constant speed

Fig. 7 Velocity triangles relating boat and wind speeds

essentially unaffected by the wave formations arising f r o m speed and leeway. Therefore, f o r a given heel angle ip:

RM^ = L X h (8)

We know RM^ and h either f r o m model tests or performance estimates, and h is assumed to be constant w i t h respect to speed and leeway. Therefore, there is one value of L for each heel angle. When viewed i n horizontal projection, the lines of action of the hydrodynamic force on the h u l l and the aerody-namic force on the huU and the aerodyaerody-namic force on the sails are usually unbalanced. T h i s unbalance indicates the yacht's weather h e l m tendency, and usually goes uncorrected i n model tests.

W i t h the e q u i h b r i u m value of L thus determined, the lee-way angle and induced drag can be f o u n d f r o m equations (4) and (7), or f r o m model test plots. T h e t o t a l resistance can then be calculated by adding the induced drag to the drag at zero side force, as i n equation (5). W i t h boat speed, resistance, l i f t , and sail area known, the Gimcrack sail coefficients f o u n d by Davidson (given i n reference [4]) yield the apparent w i n d speed and angle. F r o m these values, the true w i n d speed and course angle to the t r u e w i n d are f o u n d by the following rela-tionships (see Fig. 7):

e = VA COS(/3 -1-

i) - v.

+

The main goal i n w i n d w a r d performance testing and calcu-lations is to determine the fastest speed i n the direction of the true w i n d , or speed-made-good to w i n d w a r d VMG- Values of VT and VMG f o r a range of heel angles can be p l o t t e d as i n Fig. 2. I t is known by sailors t h a t i f a boat is "pinched," the angle to the true w i n d is improved, b u t the boat speed drops so quickly t h a t the VMG is less t h a n o p t i m u m . Similarly, i f the boat is " f o o t e d " by sailing at a wider angle, to the t r u e w i n d , boat speed improves slightly, b u t n o t enough to compensate for the wider sailing angle. T h e condition sought for o p t i m u m VMG can be represented by a polar diagram of the w i n d w a r d region (Fig. 8). T h i s figure shows the range of w i n d angles over which o p t i m u m or n e a r - o p t i m u m w i n d w a r d performance is achieved. Contours are usually p l o t t e d f o r several w i n d speeds.

Fig. 8 Contour of boat speed versus true wind angle at constant true wind speed

These principles are applied later i n the paper to include rough-water effects. For the same speed and heel angle (and therefore side force), resistance is higher, although leeway is often slightly less. T h i s results i n wider sailing angles to reach sufficient boat speed and alters b o t h the boat speed and sail-ing angle f o r best

VMG-D e s c r i p t i o n o f o c e a n w a v e s

W a v e g e n e r a t i o n b y v / i n d . T h e f a c t t h a t there is a w i n d blowing necessarily entails some degree of wave f o r m a t i o n . Factors t h a t affect t h e sea state are the strength of the w i n d , the length of t i m e t h a t the w i n d has been blowing, and the distance, or fetch, over w h i c h i t has blown. A f t e r some number of hours, the waves cease to develop f u r t h e r , a condition called a f u l l y arisen sea. Data compiled by W i l b u r Marks show t h a t the sea takes about 21/2 h r to develop f u l l y i n a 10-knot w i n d , about 10 hr i n a 20-knot w i n d , and about 23 hr i n a 30-knot wind. Extensive work on the theory of wave generation by w i n d was p e r f o r m e d by N e u m a n n d u r i n g 1948-52, and is pre-sented as A p p e n d i x B of reference [6]. For purposes herein i t is only i m p o r t a n t to realize t h a t energy f r o m the w i n d is trans-ferred to the sea to become wave energy.

Ocean waves generated by the w i n d are random i n length and height. This means t h a t an essentially i n f i n i t e combina-t i o n of wave lengcombina-ths and heighcombina-ts is presencombina-t, and combina-t h a combina-t combina-the wave p a t t e r n continually changes and never exactly repeats i t s e l f Also, the waves are short-crested. A l t h o u g h the predominant direction of the waves is w i t h the w i n d , there are components f r o m other directions t h a t cause crests to f o r m i n short lengths and then disappear. One never sees an ocean wave whose crest extends f r o m horizon to horizon, or w h i c h lasts for very long. However, i n spite of their random nature, there always seems to be some characteristic length and height to the waves. There is some sort of statistical average t h a t gives the impres-sion of a repeating p a t t e r n of waves, represented by the significant wave length and height. A numerical valuefor s i g n i f i -cant height is defined as the average of the highest waves. The significant height is i m p o r t a n t i n the study of motions, al-though average height is of greater interest f o r resistance.

M a t h e m a t i c a l r e p r e s e n t a t i o n o f waves. I t is only the form of ocean waves t h a t actually travels across the surface. Water particles have an o r b i t a l m o t i o n , and move back and f o r t h horizontally w i t h only a relatively slow surface current. L e t us leave random (irregular) waves f o r awhile and discuss regular (repeating) waves. The wave f o r m has a length L ^ . I t travels along the water surface w i t h crests passing any f i x e d reference p o i n t at a frequency

and at a speed or celerity

(10)

(11) Thus, longer components of waves t r a v e l faster t h r o u g h the water t h a n shorter ones. I n fact, the longer components of ocean waves travel faster t h a n the w i n d t h a t created t h e m .

Wave height equals twice the wave a m p l i t u d e and is inde-pendent of wave length. The r a t i o o f wave height to wave length gives a measure of t h é wave steepness. A r a t i o of hjLu, of about is f a i r l y common f o r waves i n the range of wave lengths of interest t o small vessels.

I n Fig. 9 i t is shown how two regular waves having d i f f e r e n t wave lengths and amplitudes can be added together to f o r m an irregular wave. T h i s is called linear superposition of the regular waves. The p r i n c i p l e of superposition can be extended so t h a t , given enough component regular waves, any irregular

Fig. 9 irregular wave formed by superposition of t w o regular waves

6J CJ^du,

CJ (SEC-) Fig. 10 Typical sea spectrum

waveform can be produced. N o r m a l l y 15 to 20 components are s u f f i c i e n t . Conversely, given an irregular waveform, the com-ponent regular waves can be f o u n d .

T h i s principle is used i n d e f i n i n g a sea state as the statisti-cal average of its i n f i n i t e components. T h e sea is defined i n terms of its energy, or the sum of its component wave energies. Wave energy is a f u n c t i o n of its amplitude:

(12) B y s u m m i n g the energy f o r the entire range of wave lengths, hence frequencies, present i n the sea, an energy spectrum is obtained. (See F i g . 10.) The f u n c t i o n S^{w) is called the "spec-t r a l densi"spec-ty." T h e area under "spec-the curve of spec"spec-tral densi"spec-ty versus frequency is proportional to the t o t a l wave energy per u n i t area of sea surface. I f one considers a l l waves whose frequencies are between coi and C02, the energy i n those waves equals

(13)

T h i s could be represented as a single wave having the central frequency of the band and an amplitude f o u n d f r o m equations (12) and (13):

Ju.

(14)

T h e average wave height f o r the entire spectrum can be deter-m i n e d b y i n t e g r a t i n g f r o deter-m a; = 0 to w = » . T h e average wave length corresponds t o the frequency at the centroid of the area under the spectral density curve. The spectral density of a moderate sea becomes negligible at a lower frequency o f about 0.4 rad/sec a n d an upper frequency of about 3 rad/sec, corre-sponding to wave lengths between, about 1200 and 20 f t .

The sea spectrum used later i n the paper is the PiersonM o s k o w i t z spectrum adopted by the 12th I n t e r n a t i o n a l T o w -ing T a n k Conference i n Rome, 1969:

effect on added resistance. However, i t does n o t necessarily follow t h a t a yacht t h a t pitches more has more added resis-tance. There are complications due t o the phase lag relative to the wave system and to heave, to h u l l f o r m , and to the f o r w a r d speed of the yacht.

T h e various motions and forces resulting f r o m the encoun-ter of waves are strongly time-dependent, and so i t is logical to relate t h e m to the frequency at w h i c h the yacht encounters the waves. Frequency of encounter, co^, is f o u n d by the f o l -lowing expression:

(15) 7U COSp (16)

A = 8.38 sq ft-sec<

B = 33.56(;;,/3)2

T h i s is a p o i n t spectrum and, therefore, corresponds t o a long-crested irregular sea. The spectra for several w i n d strengths are shown i n F i g . 11. As the w i n d strength increases, the ener-gy is concentrated i n longer waves, whose amplitudes are greater t h a n shorter waves. The shorter waves reach a l i m i t i n amplitude as the crests begin to blow off.

M o t i o n s a n d a d d e d r e s i s t a n c e i n w a v e s Response motions i n r e g u l a r waves. A vessel i n regular waves responds w i t h the same frequency as the wave, although her motions m i g h t lag consistently behind the m o t i o n of the wave. For example, as the wave crest passes amidships, when m a x i m u m heaving m i g h t be expected, a vessel s t i l l moves up-w a r d f o r a short t i m e before i t reverses direction and begins f a l l i n g i n t o the wave trough. I t is this lag between the passing of the wave crest and the vertical reaction of the h u l l t h a t is p r i m a r i l y responsible f o r added resistance caused by encoun-t e r i n g waves.

A l t h o u g h the study of motions is very i m p o r t a n t for ships, i t is somewhat i n c i d e n t a l for racing yachts. Speed-made-good t o w i n d w a r d is w h a t counts when beating i n t o the seas, and this is only loosely related to motions. The essential matter is t o m i n i m i z e added resistance and leeway. B u t since motions are more easily understood t h a n resistance and side force, t h e y w i l l f i r s t be discussed briefly.

The p r i n c i p a l motions o f a yacht sailing t o w i n d w a r d are p i t c h and heave. R o l l i n g is nearly eliminated by the d a m p i n g effects o f the sails, and sway and surge have insignificant fect. Yaw is i m p o r t a n t w i t h regard to control, b u t has l i t t l e ef-fect on resistance. P i t c h is more i m p o r t a n t t h a n heave i n its

2 5 0 200 ISO FT'-SZC \00 50 1 1 1 -rs

-•1

Fig. 11 Pierson-Mosi<owitz sea spectra for severai wind veiocities

0 . Ê o.e 1 . 0

C L) ( S E C - ' )

where OJ is the frequency of the wave at a stationary point, u the yacht's f o r w a r d speed, and n the angle of wave incidence, measured f r o m the direction of wave propagation t o the yacht's course. Thus, f o r f o l l o w i n g seas ju = 0 deg, and f o r head seas (It = 180 deg. W h e n calculating the response of the yacht, n o t only the motions and resistance b u t also the sea spectrum must be corrected f o r the yacht's f o r w a r d speed by relating ev-erything to frequency o f encounter. (See references [7, 8] and following section.)

I f the yacht heaved exactly i n phase w i t h the wave, the mo-t i o n of mo-the cenmo-ter of gravimo-ty o f mo-the yachmo-t w o u l d follow mo-the wave exactly. Due to the mass of the yacht, however, i t takes t i m e to respond t o the hydrodynamic forces. Only i n waves approxi-mately h a l f again as long as the yacht does the heaving m o t i o n approach the wave m o t i o n . I n very short waves, the response is so m u c h slower than the waves t h a t almost no m o t i o n exists. Between the two extremes is a condition of resonance where the amplitude o f heaving m o t i o n even exceeds the wave am-plitude. Figure 12 shows the heaving m o t i o n i n regular waves of the 12metre yacht Constellation as a r a t i o of (heave a m p l i -tude)/(wave a m p l i t u d e ) , and the corresponding phase angles. T h e phase angle is the p a r t o f an encounter cycle or period by w h i c h the m a x i m u m m o t i o n lags the passing of the wave

am-Fig. 12 Heave response amplitudes and phase angles for Constellation

idships. The heave response d i d not vary significantly over a speed range of 6V2 to knots.

P i t c h i n g is related to the wave slope, 27rf/Lu,. T h e ratio of p i t c h amplitude to wave slope, Lu, 0/27ri", behaves similarly t o heave i n t h a t i t goes f r o m nearly zero i n short waves, t h r o u g h a period of resonance, and eventually converges to a value of u n i t y i n long waves (Fig. 13). P i t c h response amplitude exhib-ited a slight dependence on speed, b u t the phase angles d i d not.

R e s i s t a n c e a n d side f o r c e i n waves. Resistance of the yacht is related to energy, and, like wave energy, the added resistance i n waves is very nearly p r o p o r t i o n a l to wave a m p l i -tude squared. There are assumed to be no Reynolds effects, and so the resistance is expanded by Froude scaling. W h e n p l o t t e d against frequency of encounter, there is seen to be a clear dependence on f o r w a r d speed (see Fig. 14).

Side force f o r a given speed and leeway is generally i n -creased by wave action. Since the conditions of e q u i l i b r i u m are such t h a t a yacht sails at essentially constant side force f o r constant heel, this means t h a t leeway is reduced i f speed is maintained. (As speed is reduced, leeway increases.)

I t was assumed t h a t the increase of lift-slope coefficient i n waves is p r o p o r t i o n a l to wave a m p l i t u d e . Thus, f o r a given speed and leeway, the side force increase would be p r o p o r t i o n -al to wave amplitude. Since the corresponding induced drag w o u l d be p r o p o r t i o n a l to (side force)^, i t would also be propor-t i o n a l propor-to (wave amplipropor-tude) 2. T h i s is consispropor-tenpropor-t, propor-then, w i propor-t h propor-the general behavior of added resistance i n waves. The increment i n drag slope has been defined as

F r o m this one obtains

Because tests were r u n at constant speed and leeway, the

30

to

7.5

y Fn- .32

Fig. 13 Pitch response ampiitudes and phase angies for Constellation

t o t a l measured added resistance was complicated by the fact t h a t i t included n o t only the usual added resistance due to waves, b u t also the added induced drag due to the additional side force. The induced drag due t o side force was separated f r o m the drag induced by wave action i n the f o l l o w i n g way. T h e induced drag slope was taken to be the same i n rough water as i n smooth water, and the induced drag corresponding t o the measured side force was calculated and deducted f r o m the t o t a l measured added resistance. T h e r e m a i n i n g added re-sistance was f o u n d t o be nearly independent o f yaw angle (hence side force), and can be considered as the added resis-tance due to waves alone. RAW- T h e v a l i d i t y o f the procedure is based on a l i m i t e d amount of data, b u t i t is adequate f o r present use.

Because stability i n smooth water is relatively unaffected by the hull's waveform, this behavior is extended t o rough water, since no test data on s t a b i l i t y i n waves are available. There-fore, side force and induced drag are taken to be the same i n smooth and rough water. Leeway, however, does change, and can be considered separately, as described i n the foregoing.

Resistance i n rough water can now be p l o t t e d i n a rather simple manner. Recalling the smooth-water procedure i n the f i r s t section, the t o t a l resistance RT = Ro + Ri is calculated f o r a range of speeds at the assumed heel angle and side force. A f t e r computing the added resistance due to waves, RAW is added to Ro + Ri io yield the total rough-water resistance. A procedure to calculate RAW follows.

i? R W = R , + R , + RMV (19)

C a l c u l a t i o n p r o c e d u r e s i n o b l i q u e i r r e g u l a r w a v e s A v e r a g e added resistance. I n m a k i n g speed predictions i n irregular waves, i t is the average t o t a l resistance f o r a given speed t h a t is required. The added resistance i n regular waves having been determined by model tests, and the transfer f u n c

-1.0 2.0 3.0 40

Fig. 14 Sea spectrum, transfer function, and response spectrum for several speeds and 14-ft gyradius

tions (RAW/^^) = | i ? A 0 | 2 p l o t t e d versus frequency of encoun-ter, the average added resistance i n irregular waves is f o u n d by calculating the resistance response spectrum and integrat-ing i t to o b t a i n its area (Fig. 15). I n general f o r m , the response spectral density

.S„(u) = \RAO\^ X Sj-(a)) (20)

W h e n there is f o r w a r d m o t i o n , must be substituted f o r co. Since the RAO's have already been given as functions o f cx>e, only the sea spectrum requires t r a n s f o r m a t i o n .

T h e t o t a l wave energy m u s t remain the same as a result of the t r a n s f o r m a t i o n . Therefore,

S,-(a.,) = S.c(co)£ (21)

St. Denis and Pierson [7] show t h a t the Jacobean t o t r a n s f o r m the sea spectrum accounting f o r b o t h frequency o f encounter and angle o f incidence i n t o oblique seas is

dcOe _{y i - 4a,} (22) where O^fV C O S M Ü1-V COSH g I 4v oosH I 0)% / I - \ ^ (23)

W i t h the sea spectral density thus transformed, the average added resistance is f o u n d by a method used by Gerritsma [9]:

(24)

(25)

Speed-magood to w i n d w a r d . T h e procedure j u s t

de-scribed w i l l now be used t o calculate the added resistance o f Constellation, based on her tests i n regular waves. B o a t speed and heading angle f o r best VMG w i l l t h e n be f o u n d and com-pared w i t h smooth-water predictions.

T h e sea state assumed i n these calculations is a Pierson-M o s k o w i t z spectrum having a 5 - f t significant wave height. T h i s has been taken t o approximate moderate coastal condi-tions i n about a 16-knot breeze. According t o the 12th I T T C ,

1969, the Pierson-Moskowitz significant height should be about 7 f t . However, the 5 - f t height was assumed because o f inaccuracies i n the Pierson-Moskowitz spectrum f o r the short-er wave lengths and heights o f intshort-erest to sailing yachts, and the frequency o f sailing i n a m e d i u m breeze w i t h less t h a n a f u l l y developed sea. T h e resulting changes i n boat speed and sailing angle are reasonably close to changes t h a t an actual 12-metre yacht would experience, so the assumed sea spec-t r u m appears spec-t o be valid f o r d e spec-t e r m i n i n g differences bespec-tween yachts under average conditions. T h i s spectrum, corrected f o r boat speeds of 6.5, 7.5, and 8.5 knots, is shown i n Fig. 14.

A l t h o u g h Constellation was tested at only 7.5 knots, her added resistance f o r a given frequency of encounter was nearly identical to t h a t of Sovereign, w h i c h was tested at a l l three speeds. I t was assumed t h a t Sovereign's added resistance RAO's at the higher and lower speeds could be used f o r Con-stellation. These are also p l o t t e d i n Fig. 14.

Using the spectrum transfer equation (24), the response spectral density Si;_4,y(a)e) is obtained by m u l t i p l y i n g the sea spectral density by the transfer f u n c t i o n . For each speed, the area under the response spectrum curve yields the average added resistance.

T h e smooth-water resistance and added resistance are added together for each of the speeds. T h e results are p l o t t e d i n Fig. 16. I t is seen t h a t the increase over the smooth water resistance is about 30 percent.

T h e next step is to read o f f values of resistance at s u f f i -ciently close speeds to determine the o p t i m u m sailing condit i o n . Usually y4knocondit inconditervals are adequacondite. T h e n condithe G i m -crack coefficients are entered w i t h the resistance and side force to determine the apparent w i n d speed and direction f o r 'each boat speed. F r o m the velocity triangles (Fig. 7) the t r u e

w i n d speed and sailing angle are calculated.

T h e liftslope increment e is determined i n the same m a n

-150 iOO 50 GOO •^00 200 5n ( t J c )

\

#-

\

83

Fig. 15 Transfer function and response spectrum for severai speeds and 12.6-ft gyradius 1 0 0 0 4-RE5. as) 500 R O U G H W A T E R R E S I S T A N C E : J L . ' 1 4 . 0 ' . 2 0 - H E E L . . " - 1 ^ / ^ ' 6.0 7.0 8.0 3.0

Fig. 16 Smooth- and rough-water resistances of Consteltation

Table 1

Smooth Water Rough Water

kyy (ft)

_...

Vs (knots) 8.10 7.85 7.89

VMG (knots) 6.74 6.31 6.34

7 (deg) 33.7 36.5 36.5

4' (deg) 5.3 4.8 4.8

ner as added resistance. T h e n leeway angle i n rough water is calculated by

L

' / ' K W = (26)

T h e condition for best speed-made-good to w i n d w a r d is f o u n d i n both smooth and rough water. Because of slight d i f -ferences i n true w i n d speed, Constellation's results have been adjusted to a true w i n d speed VT = 15.65 knots. F r o i n Table 1 i t is seen that, compared w i t h her smooth water performance, Constellation loses about 0.4-knofs VMG- T h i s is produced b y only a 0.25-knot loss i n boat speed, combined w i t h about a 3-deg wider sailing angle. T h e loss of VMG shown i n the table corresponds to about 2 m i n 45 sec on a 4.5-mile weather leg used i n America's Cup racing.

E f f e c t of gyradius on speed-made-good. I t has been argued t h a t the l o n g i t u d i n a l d i s t r i b u t i o n of weight has an i m -p o r t a n t effect on w i n d w a r d -performance. T h i s d i s t r i b u t i o n o f weight is represented by the gyradius, or the distance f r o m the yacht's center of g r a v i t y a t w h i c h all her weight w o u l d be lo-cated to have an equivalent moment of inertia. T h e m o m e n t of inertia increases as the square of the gyradius:

lyy = mky (27)

I t is the experience of many sailors t h a t reducing the gyrad-ius (concentrating weights amidships) improves a yacht's per-formance to w i n d w a r d . There are occasions, however, such as i n a short, sloppy sea, when spreading weights o u t seems t o help. A l t h o u g h this is p a r t l y due to more power f r o m slower m o t i o n and therefore less shaking of the sails, the resistance is slightly less i n short waves as well.

A n extreme change i n gyradius was made i n two other 12-metre models, and the results were proportioned t o approxi-mate such a change to Constellation. Figure 17 shows the re-sults. Whatever advantage the higher gyradius m i g h t h o l d i n waves haying a length less t h a n about 45 f t (OJ^ greater t h a n about 3.5 sec) is a poor t r a d e o f f f o r the substantial increase i n longer waves, unless unusual special conditions prevail. T h i s result is consistent w i t h the findings of Spens et aL [2].

I n the design o f actual yachts, i t is very d i f f i c u l t to alter the gyradius b y as m u c h as 10 percent. For example, i n the change o f 12-metre construction f r o m wood t o a l u m i n u m , the

gyrad-ius w i l l only be reduced by about 5 to 8 percent. As an exam-ple, the effects of a 10-percent change i n gyradius on Constel-lation have been checked. T h e transfer f u n c t i o n s and re-sponse spectra are shown i n Fig. 15. I t is apparent f r o m Figs. 15 and 16 t h a t the reduction i n added resistance is substan-tial. A t 7.5 knots i n the sea state assumed i n the foregoing, the added resistance is about 195 lb f o r a 14.0-ft gyradius and about 170 l b f o r a 12.6-ft gyradius, a reduction of about 13 percent (shown i n Pig. 16). Table 1 shows t h a t the improve-ment i n VMG is about 0.03 knots, representing about 12 sec on a 4.5-mile weather leg, or about 36 sec i n a Cup race—enough to be a significant factor.

Heave a n d pitch. I n irregular waves, one is usually con-cerned only about the larger-motion amplitudes. The signifi-cant motions, therefore, are good indicators. T h e signifisignifi-cant m o t i o n , or average of the Va-highest motions, is a m o t i o n am-p l i t u d e exceeded by an average of only one out of every six re-sponses. As a consequence of assuming a n o r m a l (Gaussian) statistical d i s t r i b u t i o n of the response spectra, the significant heave and p i t c h are determined f r o m the area under the re-sponse spectral density curve, or variance, mo, according to the f o l l o w i n g relationships:

(28) (29) Using the same sea spectra as before, the heave and p i t c h re-sponse spectra are calculated f r o m the transfer functions shown i n F i g . 18. T h e p i t c h transfer f u n c t i o n is obtained f r o m the r a t i o of p i t c h a m p l i t u d e wave slope, L„ö/2Trf = d/k f , plot-ted i n Fig. 13. Values f o r Constellation at 6.5 knots and 8.5 knots were proportioned f r o m results of Sovereign.

Calculations of Constellation's significant heave and p i t c h indicated a very small dependence on speed. A t a boat speed Vs = 7.5 knots, the significant-heave double a m p l i t u d e is 4.85 f t and the significant-pitch double a m p l i t u d e 14.6 deg.

C o n c l u s i o n s

T h e foregoing calculations demonstrate how resistance, speed, and motions can be determined on the basis of heeled model tests i n oblique regular waves. Such calculations entail s i m p l i f y i n g assumptions w h i c h are discussed i n the following.

T h e calculations of responses i n irregular waves are based on the principle of linear superposition. Superposition means

1.0

0.5

Fig. 17 Added resistances of Constellation ai two gyradii

Fig. 18 Heave and pitch response amplitude operators for Constella-tion at 7.5-knots boat speed, showing corresponding wave lengths

t h a t the t o t a l response is obtained by adding the effects of a number of component responses ( i n f i n i t e i n this analysis). L i n e a r i t y means t h a t motions are p r o p o r t i o n a l to wave height, and t h a t resistance is p r o p o r t i o n a l to (wave height) 2. T h e as-s u m p t i o n of linearity ias-s not exactly correct, b u t i f the wave amplitudes i n which the model is tested are close to those i n the assumed sea spectrum, the errors w i l l be small.

Response i n irregular waves depends upon w h a t irregular sea spectrum is assumed. A t present, since accurate, nonpro-p r i e t a r y data f o r snonpro-pectra anonpro-pnonpro-prononpro-priate to small c r a f t are nearly nonexistent, i t is necessary to c o n f i r m i f the assumed spec-t r u m is j u s spec-t i f i e d by checking whespec-ther spec-the resulspec-ts are of reason-able magnitude. T h e n the predictions of differences between yachts w i l l be f a i r l y accurate.

I n m a k i n g w i n d w a r d predictions, i t has been assumed t h a t the average s t a b i l i t y of the yacht does n o t change i n a seaway, and t h a t the Gimcrack sail coefficients apply. I t is l i k e l y t h a t the efficiency of the sails is reduced b y the yacht's motions. However, no means is presently available to correct f o r this.

F i n a l l y , because Constellation was tested at only one speed, her response characteristics were approximated f o r higher and lower speeds and f o r lower gyradius. Her responses at the m i d -dle speed were so similar to those of Sovereign, and the yachts are so similar i n shape and characteristics, t h a t such adjust-m e n t f o r speed seeadjust-ms j u s t i f i e d . The t w o other adjust-models whose gyradii were reduced were somewhat unlike each other, y e t their responses at the lower gyradius behaved i n nearly the same proportions to the higher gyradius. These proportions, therefore, were used f o r Constellation.

T h i s paper shows how one design parameter, gyradius, af-fects the speed-made-good to w i n d w a r d . The same principles can be applied t o investigate changes i n s t a b i l i t y and sailplan, and d i f f e r e n t h u l l forms can be tested and compared on an equal basis. As improvements are made i n resistance predic-t i o n mepredic-thods i n smoopredic-th and rough wapredic-ter, predic-the model predic-tespredic-t dapredic-ta described i n this paper can be supplanted by calculated esti-mates.

These and other recent rough-water test results indicate t h a t i t is permissable t o add together the various resistance components i n smooth and rough water as hypothesized b y Davidson 40 years ago. As a f i r s t approximation, i t even ap-pears t h a t induced drag i n rough water can be accounted f o r on the basis of smooth-water drag slopes. Also, changes i n lee-way resulting f r o m changes i n side force can be determined f r o m yawed tests i n waves.

Because of the substantial increase i n resistance and loss of speed of yachts i n rough water, i t is very i m p o r t a n t to consider t h e i r speed-made-good i n the anticipated sea conditions. T h e procedures i n this paper to p r e d i c t rough-water performance f r o m model tests i n regular waves can help the designer more accurately develop and select a yacht based on realistic w i n d -w a r d sailing conditions.

R e f e r e n c e s

1 Davidson, K. S. M., "Some Experimental Studies of the Sailing Yacht," Trans., SNAME, Vol. 44, 1936, pp. 288-344.

2 Spens, P. G., DeSaix, P., and Brown, P. W., "Some Further Ex-perimental Studies of the Sailing Yacht," Trans., SNAME, Vol. 75, 1967, pp. 79-111.

3 Kirkman, K. L., "Scale Experiments with the 5.5 Metre Yacht Antiope," Proceedings of the Chesapeake Sailing Yacht Symposium, Jan. 1974.

4 Spens, P. G., "Sailboat Test Technique," TM-124, Davidson Laboratory, Stevens Institute of Technology, Hoboken, New Jersey, Oct. 1966.

5 DeSaix, P., "Systematic Model Series in the Design of the Sail-ing Yacht Hull," Symposium Yacht Architecture '71, HISWA, Inter-dijk B. v., Sciphol-Oost, Holland, pp. 42-48.

6 Korvin-Kroukovsky, B. V., Theory of Seakeeping, SNAME, 1961.

7 St. Denis, M , and Pierson, W. J., Jr., "On the Motions of Ships in Confused Seas," Tra/is, SNAME, Vol. 61, 1953, pp. 280-357.

8 Lewis, E. V. and Numata, E., "Ship Motions in Oblique Seas," Trans., SNAME, Vol. 68,1960, pp. 510-547.

9 Gerritsma, J., Van der Bosch, J. J., and Beukelman, W., "Pro-pulsion in Regular and Irregular Waves," International Shipbuilding Progress, Vol. 8, No. 82, June 1961, pp. 3-15.

10 Gerritsma, J. and Moeyes, G., "The Seakeeping Performance and Steering Properties of Sailing Yachts," Symposium Yacht Archi-tecture '73, HISWA, Interdijk B. V., Schiphol-Oost, Holland, pp. 107-156.

11 Gerritsma, J., "Course Keeping Qualities and Motions in Waves of a Sailing Yacht," Proceedings of the Third AIAA Sympo-sium on the Aer/Hydronautics of Sailing, American Institute of Aero-nautics and AstroAero-nautics, Vol. 10, 1971, pp. 10-29.

12 Vossers, G., "Behaviour of Ships in Waves," Resistance, Pro-pulsion and Steering of Ships, Vol. C, The Technical Publishing Company H . Stam N . V., Haarlem, The Netherlands, 1962.

13 Lewis, E. V., "The Motion of Ships in Waves," Principles of Naval Architecture, SNAME, 1967, pp. 607-717.

14 Marks, W., "The Application of Spectral Analysis and Statis-tics to Seakeeping," SNAME T&R Bulletin 1-24, Feb. 1968.

15 K i m , C. H , "Motion of Vessels in Waves", Unpublished Lec-ture Notes, OE 221, Stevens Institute of Technology, Hoboken, New Jersey, Jan.-May 1973.